Example 30 - Chapter 11 Class 12 Three Dimensional Geometry

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Example 30 (Method 1) Find the coordinates of the point where the line through the points A (3, 4, 1) and B(5, 1, 6) crosses the XY-plane.The equation of a line passing through two points with position vectors 𝑎 ⃗ & 𝑏 ⃗ is 𝒓 ⃗ = 𝒂 ⃗ + 𝜆 (𝒃 ⃗ − 𝒂 ⃗) Given the line passes through the points (𝑏 ⃗ − 𝑎 ⃗) = (5𝑖 ̂ + 1𝑗 ̂ + 6𝑘 ̂) − (3𝑖 ̂ + 4𝑗 ̂ + 1𝑘 ̂) = 2𝑖 ̂ − 3𝑗 ̂ + 5𝑘 ̂ A (3, 4, 1) 𝑎 ⃗ = 3𝑖 ̂ + 4𝑗 ̂ + 𝑘 ̂ B (5, 1, 6) 𝑏 ⃗ = 5𝑖 ̂ + 1𝑗 ̂ + 6𝑘 ̂ ∴ 𝑟 ⃗ = (3𝑖 ̂ + 4𝑗 ̂ + 1𝑘 ̂) + 𝜆 (2𝑖 ̂ − 3𝑗 ̂ + 5𝑘 ̂) Let the coordinates of the point where the line crosses the XY plane be (x, y, 0). So, 𝑟 ⃗ = x𝑖 ̂ + y𝑗 ̂ + 0𝑘 ̂ Since point crosses the plane, it will satisfy its equation Putting (2) in (1) x𝑖 ̂ + y𝑗 ̂ + 0𝑘 ̂ = 3𝑖 ̂ + 4𝑗 ̂ + 1𝑘 ̂ + 2𝜆𝑖 ̂ − 3𝜆𝑗 ̂ + 5𝜆𝑘 ̂ x𝑖 ̂ + y𝑗 ̂ + 0𝑘 ̂ = (3 + 2𝜆)𝑖 ̂ + (4 − 3𝜆)𝑗 ̂ + (1 + 5𝜆)𝑘 ̂ Two vectors are equal if their corresponding components are equal So, Solving 0 = 1 + 5𝜆 ∴ 𝜆 = (−𝟏)/𝟓 So, x = 3 + 𝜆 = 3 + 2 × (−1)/5 = 3 − 2/5 = 13/5 & y = 4 − 3𝜆 = 4 − 3 × (−1)/5 = 4 + 3/5 = 23/5 Therefore, the required coordinates are (𝟏𝟑/𝟓,𝟐𝟑/𝟓,𝟎) Example 30 (Method 2) Find the coordinates of the point where the line through the points A (3, 4, 1) and B(5, 1, 6) crosses the XY-plane.The equation of a line passing through two points A(𝑥_1, 𝑦_1, 𝑧_1) and B(𝑥_2, 𝑦_2, 𝑧_2) is (𝒙 − 𝒙_𝟏)/(𝒙_𝟐 − 𝒙_𝟏 ) = (𝒚 − 𝒚_𝟏)/(𝒚_𝟐 − 𝒚_𝟏 ) = (𝒛 − 𝒛_𝟏)/(𝒛_𝟐 − 𝒛_𝟏 ) Given the line passes through the points So, the equation of line is (𝑥 − 3)/(5 − 3) = (𝑦 − 4)/(1 − 4) = (𝑧 − 1)/(6 − 1) A (3, 4, 1) ∴ 𝑥_1= 3, 𝑦_1= 4, 𝑧_1= 1 B(5, 1, 6) ∴ 𝑥_2 = 5, 𝑦_2= 1, 𝑧_2= 6 (𝑥 − 3)/2 = (𝑦 − 4)/(−3) = (𝑧 − 1)/5 = k So, Since, the line crosses the XY plane at (x, y, 0) z = 0 5k + 1 = 0 5k = −1 ∴ k = (−𝟏)/𝟓 So, x = 2k + 3 = 2 × (−1)/5 + 3 = 3 − 2/5 = 13/5 y = −3 × (−1)/5 + 4 = 4 + 3/5 = 23/5 Therefore, the coordinates of the required point are (𝟏𝟑/𝟓, 𝟐𝟑/𝟓, 𝟎).